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Olsen Scott. The Golden Section: Nature’s Greatest Secret (2006) [54]. This list confirms a great interest in the golden mean in 21st century science. 13. The Lecture: “The Golden Section and Modern Harmony Mathematics” By the end of the 20th century, the development of the “Fibonacci numbers theory” was widening intensively. Many new generalizations of Fibonacci numbers and the golden section had been developed [20]. Different unexpected applications of Fibonacci numbers and the golden section particularly in the oretical physics (the hyperbolic Fibonacci and Lucas functions [51, 98, 106]), computer science (Fibonacci codes and the codes of the golden proportion [20, 24, 87, 89, 102]), botany (the law of the spiral biosymmetries transforma tion [37]), and even philosophy (the law of structural harmony of systems [25]) were obtained.

618033988749894848204586834365638117720309180…. Do not be astonished by this number! Do not forget that this number is an irrational one! 62 . This surprising number, which possesses unique algebraic and geometrical properties, became an aesthetic canon of ancient Greek art and Renaissance art. 11? As is shown in [40], by using this theorem, Euclid introduced the geometric construction of the golden triangle, pentagram, and dodecahedron (these geometric figures will be discussed below). 2. The Origin of the Concept and Title of the Golden Section Authorities vary over who introduced both the concept and terminology for the golden section.

10) by τ we come to the following expression for τ: 1 τ = 1+ . 11) τ This can be represented in the following form: 1 τ −1 = . 12). It is well known that any number а has its own inverse number 1/a . 1 is an inverse num ber to 10. A traditional algorithm to get the inverse number 1/a from the initial number а consists of the division of the number 1 by the number а. In general case, this is a very complex procedure. Try, for example, to get the inverse number of a = 357821093572. This can be fulfilled only by the use of modern computer.